To take a more hard probe of the entrainment law (REF ), we checked out its capability to predict the Smagorinsky constant \(C_\mathrm {S}\) , an empirical constant used in turbulent flow models in large eddy simulations (e.g. [1]}). For the simple shear layer we match the kinematic turbulent viscosity after Smagorinsky to the viscosity trivially derived from Eq. (REF ) omitting the square root in \(\sigma _\mathrm {t}\) :
\(\nu _\mathrm {t} = (C_\mathrm {S} \Delta _\mathrm {S})^2 \left| \frac{\mathrm {d V_x}}{\mathrm {d}y} \right| \equiv \beta \mathrm {v}_\perp ^2 \left(\frac{\mathrm {d}y}{\mathrm {d V_x}}\right)^2 \left| \frac{\mathrm {d V_x}}{\mathrm {d}y} \right| \approx \beta \left(\frac{\mathrm {v}_\perp }{\Delta \mathrm {V_x}} \Delta y\right)^2 \left| \frac{\mathrm {d V_x}}{\mathrm {d}y} \right|,\) 
